Long time behavior of a tumor-immune system competition model perturbed by environmental noise
- Ying Li^{1} and
- Dongxi Li^{1}Email author
https://doi.org/10.1186/s13662-017-1112-7
© The Author(s) 2017
Received: 22 October 2016
Accepted: 7 February 2017
Published: 21 February 2017
Abstract
This paper investigates the long time behavior of tumor cells evolution in a tumor-immune system competition model perturbed by environmental noise. Sufficient conditions for extinction, stochastic persistence, and strong persistence in the mean of tumor cells are derived by constructing Lyapunov functions. The study results show that environmental noise can accelerate the extinction of tumor cells under immune surveillance of effector cells, which means that noise is favorable for the extinction of tumor in this condition. Finally, numerical simulations are introduced to support our results.
Keywords
1 Introduction
Cancer is becoming the leading cause of death around the world, but our cognition of its causes, methods of prevention and cure are still in its infancy. One great method that has shown its potential in our better understanding of such a complicated biological problem is mathematical modeling [1]. Tumor immune models have existed since the early 1990s. Researchers have proposed various modeling approaches using ordinary and delayed differential equations [2–5]. A good summary of early works of tumor immune dynamics can be found in [6]. A detailed review of non-spatial models described by ODEs is published by Eftimie [7]. A simple but classical mathematical model of a cell mediated response to a growing tumor cell is proposed and analyzed by Kuznetsov and Taylor [8]; it takes into account the penetration of tumor cells by effector cells as well as the inactivation of effector cells. Their model can be applied to describe two different mechanisms of the tumor: tumor dormancy and sneaking through. Galach [9] firstly simplified the Kuznetsov-Taylor model by replacing the Michaelis-Menten form with a Lotka-Volterra form for the immune reactions. Then time delay was considered in the simplified model, and a state of the returning tumor was observed. More complete bibliography about the evolution of a cell and the relevant role of cellular phenomena in directing the body toward recovery or toward illness can be found in [10, 11]. The detailed descriptions of virus, antivirus, and body dynamics are available in [12–15].
Conventional treatments of cancer, such as radiotherapy and chemotherapy, usually act on the tumor cells themselves, and although these conventional treatments respond early, recurrences and drug resistance often occur late in the course of long-term treatment. Cancer immunotherapy has recently gained exciting progress. In contrast to conventional therapies, immunotherapy elicits immune system antitumor responses by acting on the immune system. Immune-Checkpoint Inhibitors clinical trials have shown a greater sustained response than conventional chemotherapy. The positive response of immunotherapy generally depends on the interaction of tumor cells and the immune regulation in the tumor microenvironment. In fact, tumor microenvironment is inevitably affected by environmental noise, which is an important component in realism. Nowadays, noise dynamics have been widely studied in different fields such as epidemic model [16], nervous system [17], genetic regulatory system [18–20], bistable system [21–23], chaotic system [24, 25], etc. In the last years, stochastic growth models for cancer cells were studied in [26–30], in which Lyapunov exponent method and Fokker-Planck equation method are used to investigate the stability of the stochastic model by numerical simulations. The goal of this paper is to explore the long time behavior of tumor and effector cells in the tumor-immune system competition model perturbed by environmental noise. One of many advantages of our paper is that we initially make use of the methods of Itô’s formula and Lyapunov function to derive and analyze the properties of a stochastic tumor-immune system competition model. The other advantage of this paper is that the conditions for extinction, stochastic persistence, and strong persistence in the mean of tumor cells are established by strict mathematical proofs. Accordingly, the sufficient conditions for extinction and persistence could provide us more effective and precise therapeutic schedule to eliminate tumor cells and improve the treatment of cancer.
This paper is organized as follows. In Section 2, we introduce a mathematical model. In Section 3, we establish the sufficient conditions for extinction, stochastic persistence, and strong persistence in the mean of tumor cells. Numerical simulations are presented in Section 4, and they are used to verify and illustrate the theorems of Section 3. In Section 5, we discuss the conclusions and future directions of the research.
2 Mathematical model
When unknown tissues, organisms, or tumor cells appear in a body, the immune system tries to identify them and, if it succeeds, it attempts to eliminate them. The immune system response consists of two distinct interacting responses: the cellular response and the humoral response. The cellular response is carried by T lymphocytes. The humoral response is mediated by B lymphocytes. The dynamics of the antitumor immune response in vivo is complicated and not well understood.
Once the tumor cells are identified, the immune response begins. Then tumor cells are caught by macrophages, which can be found in all tissues in the body and circulate in the blood stream. Macrophages absorb tumor cells, eat them, and release a series of cytokines which activate T helper cells (i.e., a subpopulation of T lymphocytes). Activated T helper cells coordinate the counterattack. T helper cells can also be directly stimulated to interact with antigens. These helper cells cannot kill tumor cells, but they send urgent biochemical signals to a special type of T lymphocytes called natural killers (NKs). T cells begin to multiply and release other cytokines that further stimulate more T cells, B cells, and NK cells. As the number of B cells increases, T helper cells send a signal to start the production of antibodies. Antibodies circulate in the blood and are attached to tumor cells, which implies that the tumor cells are more quickly engulfed by macrophages or killed by NK cells. Like all T cells, NK cells are programmed to identify one specific type of infected cell or cancer cell. NK cells are lethal and constitute a vital line of the defense.
3 Long time behavior of the stochastic model
- (1)
The tumor cells \(y(t)\) will go to extinction a.s. if \(\lim_{t\rightarrow +\infty }y(t)=0\).
- (2)
The tumor cells \(y(t)\) will be stochastically permanent a.s. if there are constants \(N > 0\) and \(M > 0\) such that \(\mathcal{P}_{ \ast }\{y(t)\geq N\}\geq 1-\xi \) and \(\mathcal{P}_{\ast }\{y(t) \leq M\}\geq 1-\xi\).
- (3)
The tumor cells \(y(t)\) will be strong persistent in the mean a.s. if \({\langle y(t)\rangle }_{\ast }>0\).
Next we establish the sufficient conditions of extinction and persistence for our model.
Lemma 1
Proof
- (i)When \(0<\alpha -x\) and \(\alpha \beta y_{0}\leq \alpha -x\) (i.e., \(0< x\leq \alpha (1-\beta y_{0})\)),$$\begin{aligned} y =&\frac{1}{{{\frac{\alpha \beta }{\alpha -x}+(\frac{1}{y_{0}}-\frac{ \alpha \beta }{\alpha -x}})e^{-(\alpha -x)t}}} \\ =&\frac{1}{\frac{\alpha \beta }{\alpha -x}+\frac{\alpha \beta }{ \alpha -x}(\frac{\alpha -x}{\alpha \beta y_{0}}-1)e^{-(\alpha -x)t}} \\ \leq& \frac{1}{\frac{\alpha \beta }{\alpha -x}} \\ \leq& \frac{1}{ \beta }. \end{aligned}$$
- (ii)When \(0<\alpha -x<\alpha \beta y_{0}\) (i.e., \(\alpha (1-\beta y_{0})< x<\alpha \)),$$\begin{aligned} y =&\frac{\alpha -x}{\alpha \beta +(\frac{\alpha -x}{y_{0}}-\alpha \beta)e^{-(\alpha -x)t}} \\ < &\frac{\alpha -x}{{\alpha \beta +(\frac{ \alpha -x}{y_{0}}-\alpha \beta)}}=y_{0}. \end{aligned}$$
- (iii)When \(\alpha -x\leq 0\) (i.e., \(x\geq \alpha \)),$$\begin{aligned} y &=\frac{x-\alpha }{-\alpha \beta +(\alpha \beta -\frac{\alpha -x}{y _{0}})e^{-(\alpha -x)t}} \\ &=\frac{x-\alpha }{-\alpha \beta +(\alpha \beta +\frac{x-\alpha }{y _{0}})e^{-(\alpha -x)t}} \\ &\leq \frac{x-\alpha }{-\alpha \beta + \alpha \beta +\frac{x-\alpha }{y_{0}}}=y_{0}. \end{aligned}$$
Lemma 2
Proof
Theorem 1
For any positive initial value \((x_{0},y_{0})\), particularly, \(x_{0}<\frac{1}{\beta }\), Equation (4) has a positive unique global solution \((x(t),y(t))\) on \(t\geq 0\) a.s.
Proof
Remark 1
In order to guarantee the existence and uniqueness of the solution of model (4), we discuss the extinction and persistence of \(y(t)\) under the condition \(x_{0}<\frac{1}{\beta }\) below.
Theorem 2
Proof
Theorem 3
If \(\alpha \delta -\frac{\omega \alpha }{4\beta }>\varepsilon \), then the tumor cells \(y(t)\) will be stochastically permanent almost surely.
Proof
Theorem 4
If \(\delta \alpha >\varepsilon \), then the tumor cells \(y(t)\) will be strongly persistent in the mean almost surely.
Proof
4 Numerical simulations
In this section, we use the Euler-Maruyama numerical method mentioned by Higham [37] to support our results.
5 Conclusion
- (A)
If \(\alpha \delta <\varepsilon \), then the effector cells \(x(t)\) go to \(\frac{\varepsilon }{\delta }\), and the tumor cells \(y(t)\) will go to extinction a.s.
- (B)
If \(\alpha \delta -\frac{\omega \alpha }{4\beta }>\varepsilon \), then the tumor cells \(y(t)\) will be stochastically permanent a.s.
- (C)
If \(\alpha \delta >\varepsilon \), then the tumor cells \(y(t)\) will be strongly persistent in the mean a.s.
Our work reveals some important and interesting biological results. By comparing the results of Galach [9] with our work of Theorem 2, we can see that environmental noise can change the properties of tumor-immune population dynamics significantly. For example, under the same conditions, Figure 4(b) shows that the mean time to extinction of the stochastic model is less than the time in the deterministic model. Namely, environmental noise can accelerate the extinction of tumor cells under the surveillance of effector cells, which means that noise is favorable for the extinction of tumor in this condition.
Some interesting questions deserve further investigation. For example, in our model, we assume that fluctuations in the environment will mainly affect the immune coefficient. It is interesting to study what happens if noise affects other parameters of the system. Another question of interest is to consider the stability in distribution (e.g., [38, 39]). Moreover, one may propose some realistic but complex models. An example is to add the treatment into the system. The motivation is to predict the effect of treatment and design the optimal treatment schedule.
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11402157 and 11571009), Shanxi Scholarship Council of China (Grant No. 2015-032), Technological Innovation Programs of Higher Education Institutions in Shanxi (Grant No. 2015121) and Applied Basic Research Programs of Shanxi Province (Grant No. 2016021013).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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